Regression 2
Regression 2 : The statistical counterpart or analogue of the functional expression, in ordinary mathematics, of one variable in terms of others. A random variable is seldom uniquely determined by any other variables, but it may assume a unique mean value for a prescribed set of values of any other variables. The variate y is statistically dependent upon other variates xy, x2, . . . , xn when it has different probability distributions for different sets of values of the x's. In that case its mean value, called its conditional mean, corresponding to given values of the x's will ordinarily be a function of the x's. The regression function Y of y with respect to X], x2, . . . , xK is the functional expression, in terms of the x's of the conditional mean of y. This is the basis of statistical estimation or prediction of y from known values of the x's.From the definition of the regression function, we may deduce the following fundamental properties: E(Y) = E(y), E(y-y) = 0; E[Y(y -Y) = 0, ?(y2) = E(yY); a2(y) = a2(y) = a2(Y) + a2(y-Y), where ahv denotes the variance of any variate w, and ?(w) denotes the expected value of w.The variate y is called the regressand, and the associated variates x,, x2, . . . , x" are called regressors; or alternatively, y is called the predictand and the x's are called predictors. When it is necessary to resort to an approximation Y' of the true regression function Y, the approximating function is usually expanded as a series of terms Ylt Y2, . . . , Ym each of which may involve one or more of the basic variates xp x2, . . . , xK. By extension of the original definitions, the component functions Yv Y2, . . . , Ym are then Also called: regressors or predictors. Various quantities associated with regression are referred to by the following technical terms: The variance rj2(y) of the regressand is called the total variance. The quantity y - Y is variously termed the residual, the error, the error of estimate. Its variance a2(y - 10 is called the unexplained variance, the residual variance, the mean-square error; and its positive square root
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